Subgroups — #13322945

Order 1792.

730 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#89696	C₂	1	2	yes	maximal
3	#11728504	C₂	1	8	yes	maximal, complemented
4	#11727485	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂²×#1136	—	2	2	no	—
7	C₇×#431	C₄	1	4	yes	—
8	C₇×#431	C₂²	1	4	yes	—
9	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
12	#89696	—	2	2	no	—
13	#89696	—	2	2	no	—
14	#89696	C₂²	1	4	yes	—
15	#89696	C₂²	1	4	yes	—
16	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
17	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
18	#89253	C₂²	1	16	yes	complemented
19	#17814	—	7	1	no	maximal, Sylow
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	D₄	1	2	yes	—
27	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₂×#1136	—	2	4	no	—
33	C₂×#1136	—	2	4	no	—
34	C₂×#1136	—	4	8	no	—
35	C₂²×C₇⋊C₈	—	2	4	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
37	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
38	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
39	C₄×C₇×C₈	C₂³	1	4	yes	—
40	C₄⋊(C₇⋊C₈)	—	2	4	no	—
41	C₄⋊(C₇⋊C₈)	—	2	4	no	—
42	C₄⋊(C₇⋊C₈)	—	2	4	no	—
43	C₄⋊(C₇⋊C₈)	C₂³	1	8	yes	—
44	C₂×#431	D₇	1	1	yes	characteristic
45	C₂×#423	—	7	2	no	—
46	#1571	—	7	8	no	—
47	#1567	—	7	4	no	—
48	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	4	2	no	—
50	C₂²×C₄×C₇	—	4	2	no	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	D₈	1	4	yes	—
58	C₂²×C₄×C₇	Q₁₆	1	4	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
65	C₂×C₇×C₈	—	4	4	no	—
66	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
67	C₄²×C₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₇	C₂×D₄	1	2	yes	—
73	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇	C₂×Q₈	1	2	yes	—
75	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₇	C₂⁴	1	1	yes	characteristic
77	#1136	—	4	8	no	—
78	#1136	—	4	8	no	—
79	C₂×C₇⋊C₈	—	2	8	no	—
80	C₂×C₇⋊C₈	—	4	8	no	—
81	C₂²×M₁₆	—	14	2	no	—
82	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
83	#431	Dic₇	1	4	yes	—
84	#431	C₂×D₇	1	4	yes	—
85	#423	—	7	4	no	—
86	#423	—	7	4	no	—
87	#423	—	14	2	no	—
88	#423	—	14	2	no	—
89	C₂×C₄⋊C₈	—	7	4	no	—
90	C₂×C₄⋊C₈	—	14	2	no	—
91	C₂×C₄⋊C₈	Dic₇	1	2	yes	—
92	C₂×C₂²⋊C₈	Dic₇	1	2	yes	—
93	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
94	#335	—	7	16	no	—
95	C₂³×C₇	—	4	1	no	—
96	C₂³×C₇	—	4	1	no	—
97	C₂³×C₇	#107	1	4	yes	—
98	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	4	2	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	4	1	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	#107	1	4	yes	—
109	C₂×C₄×C₇	#107	1	8	yes	—
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
117	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
123	C₇×C₈	—	2	8	no	—
124	C₇⋊C₈	—	4	8	no	—
125	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
126	C₂³×C₄	D₂₈	1	1	yes	characteristic
127	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
128	C₂×M₁₆	—	14	4	no	—
129	C₂×M₁₆	—	14	4	no	—
130	C₂×M₁₆	—	28	8	no	—
131	C₂²×C₈	—	14	4	no	—
132	C₂²×C₈	C₂×Dic₇	1	4	yes	—
133	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
134	C₂×C₄²	Dic₁₄	1	2	yes	—
135	C₂×C₄²	D₂₈	1	2	yes	—
136	C₂×C₄²	D₂₈	1	1	yes	characteristic
137	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
138	C₂×C₄²	C₂×Dic₇	1	2	yes	—
139	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₇	1	2	yes	—
141	C₄⋊C₈	—	7	8	no	—
142	C₄⋊C₈	—	14	4	no	—
143	C₄⋊C₈	—	14	4	no	—
144	C₄⋊C₈	—	14	4	no	—
145	C₄⋊C₈	C₂×Dic₇	1	8	yes	—
146	C₂²⋊C₈	C₂×Dic₇	1	8	yes	—
147	C₄×C₈	C₂×Dic₇	1	4	yes	—
148	C₄×C₈	C₂²×D₇	1	4	yes	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	—	2	2	no	—
151	C₂²×C₇	—	2	2	no	—
152	C₂²×C₇	—	2	1	no	—
153	C₂²×C₇	—	4	4	no	—
154	C₂²×C₇	C₂×#107	1	4	yes	—
155	C₂²×C₇	C₂×#107	1	2	yes	—
156	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	2	1	no	—
161	C₄×C₇	C₂×#107	1	4	yes	—
162	C₄×C₇	C₂×#107	1	2	yes	—
163	C₄×C₇	#430	1	2	yes	—
164	C₄×C₇	#568	1	2	yes	—
165	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
167	C₂⁴	C₄⋊Dic₇	1	1	yes	characteristic
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	D₅₆	1	4	yes	—
171	C₂²×C₄	Dic₂₈	1	4	yes	—
172	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
173	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
178	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
180	C₂²×C₄	C₂×D₂₈	1	2	yes	—
181	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
184	M₁₆	—	28	8	no	—
185	M₁₆	—	28	8	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	—	14	8	no	—
188	C₂×C₈	—	28	8	no	—
189	C₂×C₈	C₂²×Dic₇	1	8	yes	—
190	C₄²	C₄⋊Dic₇	1	2	yes	—
191	C₄²	C₄⋊Dic₇	1	2	yes	—
192	C₄²	C₄⋊Dic₇	1	2	yes	—
193	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₄	1	2	yes	—
195	C₄²	C₂×D₂₈	1	2	yes	—
196	C₄²	C₂×D₂₈	1	1	yes	characteristic
197	C₄²	C₂²×Dic₇	1	1	yes	characteristic
198	C₄²	C₂²×Dic₇	1	2	yes	—
199	C₄²	C₂³×D₇	1	1	yes	characteristic
200	C₂×C₇	—	2	2	no	—
201	C₂×C₇	—	2	4	no	—
202	C₂×C₇	#2905	1	1	yes	characteristic
203	C₂×C₇	C₂²×#107	1	1	yes	characteristic
204	C₂×C₇	C₂×#430	1	1	yes	characteristic
205	C₂³	—	4	1	no	—
206	C₂³	—	4	1	no	—
207	C₂³	#6392	1	4	yes	—
208	C₂³	C₂×C₄⋊Dic₇	1	2	yes	—
209	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	#6392	1	4	yes	—
219	C₂×C₄	#6392	1	8	yes	—
220	C₂×C₄	C₂×D₅₆	1	4	yes	—
221	C₂×C₄	C₂×D₅₆	1	2	yes	—
222	C₂×C₄	C₂×Dic₂₈	1	4	yes	—
223	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
225	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
227	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
232	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
233	C₈	—	2	8	no	—
234	C₈	—	28	8	no	—
235	C₇	#17814	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×#6392	1	4	yes	—
242	C₂²	C₂×#6392	1	2	yes	—
243	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	C₂×#6392	1	4	yes	—
249	C₄	C₂×#6392	1	2	yes	—
250	C₄	#89897	1	2	yes	—
251	C₄	C₂²×D₅₆	1	1	yes	characteristic
252	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
253	C₄	#90510	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	2	4	no	—
256	C₂	#11739649	1	1	yes	characteristic
257	C₂	C₂²×#6392	1	1	yes	characteristic
258	C₂	C₂×#89897	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#431	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×#431	D₇
hypocenter	C₇	#17814
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #11728504	C₂	semidirect
4. #11727485	C₂	semidirect
18. #89253	C₂²	semidirect
235. C₇	#17814	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#431	2
C₂×#89696	2
#11728504	2
#11727485	2
#17814	7